Inorganic phosphorous is a naturally occurring element in all plants and animals

Question

Inorganic phosphorous is a naturally occurring element in all plants and animals, with concentrations increasing progressively up the food chain. Geochemical surveys take soil samples to determine phosphorous content in ppm.Let m1be the mean phosphorous content for region 1, and m2be the mean phosphorous content for region 2.    Based on this sample data, is there evidence that the two means are different? What is the test statistic and p-value?

X1 : phosphorous content for region 1, in ppm:

   540    810    790    790    340    800    890    860    820    640    970    720

X2 : phosphorous content for region 2, in ppm:

  750   870   700   810   965   350   895   850   635   955   710   890   520   650   280   993

Explanation / Answer

Here we have to test the hypothesis that,

H0 : m1 = m2 Vs H1 : m1 m2

where, m1 be the mean phosphorous content for region 1,

and m2 be the mean phosphorous content for region 2.

Assume alpha = level of significance = 5% = 0.05

Here n1 = 12

and n2 = 16

Both the sample sizes are small and population standard deviation is unknown so we use two sample t-test.

But before that we have to test whether variances are equal or not.

FOr variance testing the hypothesis is,

H0 : Variances are equal.

H1 : Variances are not equal.

Both testing we can done using EXCEL.

steps :

Enter all the data in EXCEL sheet --> Data --> Data Analysis --> F-Test Two - Sample for Variances --> ok --> Variable1Range : select X1 range --> Variable2Range : select X2 range --> Alpha : 0.05 --> Output Range : select one empty cell --> ok

Output is :

The test statistic F = 0.6452

P-value = 0.234

P-value > alpha

Accept H0 at 5% level of significance.

Conclusion : Variances are equal.

So we use pooled variances (equal variances).

Now two sample t-test assuming equal variances by using EXCEL is,

steps :

Data --> Data analysis --> t-test : Two-Sample Assuming Equal Variances --> ok --> Variable1Range : select X1 range --> Variable2Range : select X2 range --> Hypothesized Mean Difference : 0 --> Alpha : 0.05 --> Output Range : select one empty cell --> ok

Output :

Test statistic t = 0.1146

P-value = 0.9096

P-value > alpha

Accept H0 at 5% level of significance.

Conclusion : There is not sufficient evidence to say that the two means are different.

F-Test Two-Sample for Variances Variable 1 Variable 2 Mean 747.5 738.9375 Variance 29038.64 45006.06 Observations 12 16 df 11 15 F 0.645216 P(F<=f) one-tail 0.234159 F Critical one-tail 0.367831

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